Hi. I am Rob. I am another expert here at JA. I will be happy to assist you without a remote connetion. First let me explain the message and then give you a possible cause and solution.
As you suspect, that is a message that gmail uses when you send too many messages in a day.
The limit is between 500 and 2000 depending on your account type.
there are other limits for auto forwarded messages and messages forwarded by a rule. The only one that would be lower is that you cant have more than 20 rules auto forwarding messages. The complete list is on this page
If you have not sent many messages. than it is possible that somebody else has hacked your email address to send spam. Or a virus is using your computer to do so.
You might log into www.gmail.com and check the sent folder to see if there are messages that you did not send. But they might not show up if they used their own program to access the gmail server. Spammers are always looking for new accounts to send from. They can get your password ***** legitimate looking emails or offers that ask for your email and password ***** sometimes just through brute force cracking of a simple password.
In any case my suggestion would be to change the password ***** your gmail account. You would then have to change the apple mail program to the new password ***** well.
Then wait a day and see if it clears up.
Though mac's are not very susceptible to virus, you might also run a virus check on the computer just for safety sake.